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Equivariant Elliptic Cohomology and Mapping Stacks

This paper introduces elliptic Hochschild homology as a new cohomology theory for stacks and demonstrates that its periodic cyclic version recovers Grojnowski's equivariant elliptic cohomology for quotient stacks over the complex numbers.

Original authors: Nicolò Sibilla, Paolo Tomasini

Published 2026-06-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Nicolò Sibilla, Paolo Tomasini

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand the shape of a complex, twisting object. In mathematics, there are different "lenses" or tools to look at these shapes. One famous tool is called K-theory, which is like a way of counting the holes and loops in an object.

This paper introduces a new, more advanced tool called Elliptic Cohomology. You can think of this as a "super-lens" that sees even more intricate details than K-theory. However, this super-lens is notoriously difficult to use because it requires understanding the object not just in its current state, but in relation to an elliptic curve (a fancy, doughnut-shaped mathematical object).

Here is the paper's story, broken down into simple concepts:

1. The Problem: Too Many Maps

Usually, to study a shape (let's call it $X$ ) using an elliptic curve (let's call it $E$ ), mathematicians look at all possible ways you can wrap the doughnut $E$ around the shape $X$ .

The Issue: There are infinitely many ways to do this, including wild, tangled, topologically impossible wrappings. It's like trying to study a city by looking at every possible way a rubber band could be thrown over it, including ones that get stuck on buildings or wrap around the whole world. This creates too much "noise" to see the actual structure.

2. The Solution: "Quasi-Constant" Maps

The authors, Sibilla and Tommasini, propose a clever filter. They say, "Let's ignore the wild, tangled rubber bands. Let's only look at the quasi-constant maps."

The Analogy: Imagine you are throwing rubber bands at a city. A "constant" map is a rubber band that just sits on one spot. A "quasi-constant" map is a rubber band that is almost sitting on one spot—it might wiggle a tiny bit, but it doesn't wrap around the whole city or get stuck on a skyscraper.
The Result: By filtering out the crazy, topologically complex maps and keeping only the "almost constant" ones, the math suddenly becomes manageable. The authors prove that these specific maps behave nicely: if you break the city into neighborhoods, the maps on the whole city are just the sum of the maps on the neighborhoods. This is a crucial property that allows them to build a consistent theory.

3. The New Tool: Elliptic Hochschild Homology

Using these filtered maps, the authors define a new mathematical object they call Elliptic Hochschild Homology.

What is it? Think of it as a "function library" for the shape $X$ . Instead of just listing numbers (like a standard census), this library contains complex functions that describe how the shape interacts with the elliptic curve doughnut.
Why "Hochschild"? In math, "Hochschild homology" is a standard tool for studying shapes. The authors are essentially saying, "We are taking that standard tool and upgrading it to work with elliptic curves instead of simple circles."

4. The Big Discovery: Matching the Gold Standard

For a long time, there was a "gold standard" way to do this kind of math, developed by a mathematician named Grojnowski. It worked beautifully for complex shapes but was defined in a very specific, somewhat abstract way.

The Breakthrough: The authors prove that their new "Elliptic Hochschild Homology" is actually the same thing as Grojnowski's gold standard, provided you apply one final step.
The "Tate" Step: Imagine you have a spinning top (the mathematical object). It has a natural rotation. To get the final answer, you have to look at the top while it's spinning and take a "snapshot" of what stays the same (the fixed points). The authors show that if you take their new tool and apply this "spinning snapshot" (called the Tate construction), you get exactly the same result as Grojnowski's method.

5. Why This Matters (According to the Paper)

Geometric Clarity: The authors' method is more "geometric." Instead of just defining the math abstractly, they describe it using actual maps between shapes (the doughnut and the city). This gives a clearer picture of why the math works.
Symmetry: They show how to handle shapes that have symmetries (like a snowflake that looks the same if you rotate it). Their tool works perfectly for these "equivariant" cases.
Universality: They prove this works not just for simple shapes, but for complex "toric varieties" (shapes built from cones and polygons) and even for shapes acted upon by large, complex groups of symmetries (reductive groups).

Summary

In short, the paper says:
"We found a way to simplify the messy problem of studying shapes with elliptic curves by ignoring the 'wild' maps and only looking at the 'almost constant' ones. This creates a new, clean mathematical tool. We then proved that if you spin this tool and take a snapshot, it perfectly matches the most famous existing method for this type of math. This gives us a new, more intuitive way to understand these complex mathematical landscapes."

Technical Summary: Equivariant Elliptic Cohomology and Mapping Stacks

Problem Statement
The paper addresses the construction of equivariant elliptic cohomology for complex algebraic varieties equipped with the action of an algebraic group $G$ . While rationalized equivariant elliptic cohomology was previously constructed by Grojnowski [26] as coherent sheaves over the elliptic curve (or a product thereof), the authors seek a new geometric interpretation rooted in derived algebraic geometry. Specifically, they aim to describe this cohomology theory in terms of functions on a specific substack of the mapping stack $\text{Map}(E, [X/G])$ , where $E$ is an elliptic curve and $[X/G]$ is the quotient stack. The challenge lies in the fact that the full mapping stack is generally too large due to topologically non-trivial maps; the theory must isolate the "quasi-constant" maps that contribute to the cohomology.

Methodology
The authors employ the language of $\infty$ -categories and derived algebraic geometry, working over a fixed ground field $k$ of characteristic zero. Their approach proceeds through several key technical steps:

Quasi-Constant Maps: They introduce the derived stack of quasi-constant maps, denoted $\text{Map}_0(E, [X/G])$ . This is defined as the union of connected components of the mapping stack containing the trivial maps (those factoring through $\text{Spec}(k)$ ). Crucially, they establish that this stack satisfies a form of Zariski codescent on the target $X$ (Theorem 3.14), a property that fails for the full mapping stack.
Elliptic Hochschild Homology: Using the structure morphism $p: \text{Map}_0(E, [X/T]) \to E_T$ (where $E_T$ is the coarse moduli space of degree 0 $T$ -bundles on $E$ ), they define the $T$ -equivariant elliptic Hochschild homology of $X$ as the pushforward of the structure sheaf:
$\text{HHE}([X/T]) := p_* \mathcal{O}_{\text{Map}_0(E, [X/T])} \in \text{QCoh}(E_T)$
Local Structure and Localization: The authors analyze the local behavior of this theory. Using Luna's slice theorem and the localization of quasi-constant maps, they prove that near a point $e \in E_T$ , the stack of quasi-constant maps is equivalent to the stack of maps from the derived fixed locus $X^{T(e)}$ (Theorem 5.2). Consequently, the completion of $\text{HHE}([X/T])$ at a point $e$ is equivalent to the completion of the ordinary Hochschild homology of the fixed locus (Theorem 5.13).
Periodic Cyclic Variant and Tate Construction: Recognizing that elliptic Hochschild homology is not homotopy invariant, the authors introduce a periodic cyclic version, $\text{HPE}([X/T])$ . This is constructed by passing to the Tate fixed points of the natural $E$ -action on the mapping stack. In characteristic zero, the formal cohomology of $E$ is equivalent to that of $S^1$ , allowing the $E$ -action to be reinterpreted as a hidden $S^1$ -action. The Tate construction effectively periodizes the theory, collapsing the $\mathbb{Z}$ -grading to a $\mathbb{Z}_2$ -grading.
Adelic Descent: To establish a global comparison between their construction and Grojnowski's theory, the authors utilize adelic descent theory (following Parshin, Beilinson, and Groechenig). They show that the equivalence of the two theories can be established by verifying it on the adelic decomposition of the sheaves, which involves checking compatibility at closed points (via localization) and generic points (via the HKR theorem).

Key Results

Theorem A (Codescent): The stack of quasi-constant maps $\text{Map}_0(E, [X/T])$ satisfies $T$ -equivariant Zariski codescent. This allows the computation of elliptic Hochschild homology via local data on affine covers.
Theorem D (Toric Case): For a smooth toric variety $X$ with a maximal torus action over $\mathbb{C}$ , the elliptic Hochschild homology $\text{HHE}([X/T])$ is isomorphic to the degree 0 part of the complexified equivariant elliptic cohomology of the analytification $X^{\text{an}}$ .
Theorem G (Comparison for Tori): For a smooth quasi-projective variety $X$ with a torus action over $\mathbb{C}$ , the elliptic periodic cyclic homology $\text{HPE}([X/T])$ is isomorphic to Grojnowski's complexified equivariant elliptic cohomology $\text{Ell}_T^{\text{an}}(X^{\text{an}})$ as $\mathbb{Z}_2$ -periodic perfect complexes on $E$ .
Theorem H (Comparison for Reductive Groups): Extending the result to a general reductive group $G$ , the authors define $\text{HPE}([X/G])$ via Weyl group invariants of the torus case and prove it is isomorphic to $\text{Ell}_G^{\text{an}}(X^{\text{an}})$ .
Intrinsic Definition: The paper proposes an intrinsic definition of $G$ -equivariant elliptic Hochschild homology for reductive groups based on semi-stable quasi-constant maps to $[X/G]$ , independent of the choice of a maximal torus.

Significance and Claims
The authors claim that their work provides a geometric explanation of Grojnowski's construction through the deformation theory of quasi-constant maps from elliptic curves. By framing equivariant elliptic cohomology as a variant of Hochschild homology (specifically, functions on a derived loop space analogue), they bridge the gap between the algebraic geometry of mapping stacks and the topological origins of elliptic cohomology.

The paper emphasizes that while the construction is valid over any field of characteristic zero, the comparison with Grojnowski's theory (which relies on complex analytic geometry) requires setting $k = \mathbb{C}$ and passing to Tate fixed points. The authors note that their definition of elliptic Hochschild homology is compatible with the earlier definition by Moulinos-Robalo-Toën [48] in the non-equivariant setting, justifying their terminology.

The work is presented as a step toward a broader project aiming to provide geometric representatives for elliptic cocycles. The authors modestly state that while they establish the comparison with Grojnowski's theory and propose an intrinsic definition for reductive groups, a complete proof of the equivalence between the intrinsic and Weyl-invariant definitions for general reductive groups is reserved for future work. They also highlight that, unlike ordinary Hochschild homology, their theory is not an invariant of the derived category of perfect complexes $\text{Perf}([X/G])$ , suggesting it encodes higher categorical information.